6. If the average (arithmetic mean) of $$ b $$ and 2 is equal to the average of $$ b, 3 $$, and 4 , what is the value of $$ b $$ ? (A) 2 (B) 4 (C) 8 (D) 10 (E) 16

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UpStudy AI · Accepted Answer

We are given that the average of $$ b $$ and $$ 2 $$ is equal to the average of $$ b,\, 3,\, $$ and $$ 4 $$.

1. The average of $$ b $$ and $$ 2 $$ is:
   $$
   \frac{b + 2}{2}
   $$

2. The average of $$ b,\, 3,\, 4 $$ is:
   $$
   \frac{b + 3 + 4}{3} = \frac{b + 7}{3}
   $$

3. Set the two averages equal to each other:
   $$
   \frac{b + 2}{2} = \frac{b + 7}{3}
   $$

4. To eliminate the denominators, multiply both sides by $$ 6 $$ (the least common multiple of $$ 2 $$ and $$ 3 $$):
   $$
   6 \cdot \frac{b + 2}{2} = 6 \cdot \frac{b + 7}{3}
   $$
   This simplifies to:
   $$
   3(b+2) = 2(b+7)
   $$

5. Distribute on both sides:
   $$
   3b + 6 = 2b + 14
   $$

6. Subtract $$ 2b $$ from both sides:
   $$
   b + 6 = 14
   $$

7. Subtract $$ 6 $$ from both sides:
   $$
   b = 8
   $$

Thus, the value of $$ b $$ is $$ 8 $$.
$$ b = 8 $$

If the average (arithmetic mean) of and 2 is
equal to the average of , and 4 , what is the
value of ?
(A) 2
(B) 4
© 8
(D) 10
(E) 16

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If the average (arithmetic mean) of and 2 is equal to the average of , and 4 , what is the value of ? (A) 2 (B) 4 © 8 (D) 10 (E) 16